Volume 16, Issue 1, March 1972
Index of content:

Hydrodynamic Interaction Effects in Rigid Dumbbell Suspensions. II. Computations for Steady Shear Flow
View Description Hide DescriptionThe angular distribution of solute molecules, and the space‐averaged stresses in the fluid, are computed for steady shear flows of solutions of rigid dumbbells. The computation is done by Galerkin's method, with spherical harmonics as the trial functions. The stresses are tabulated for the full range of shear rates, and dumbbell width/length ratios, The stress components become power functions of shear rate when the latter is large. The analogy of Cox and Merz bolds better for small width/length ratios than for large ones.

The Bulk Response of Viscoelastic Solids
View Description Hide DescriptionHere the dilatational response of polymeric solids is considered. Such response is an important factor in the description of many dynamic situations including shock and acceleration wave propagation. Since such materials exhibit both instantaneous and equilibrium elasticity, as well as rate‐effects, it is assumed that polymers can be characterized as finite linear viscoelastic solids and their dilatational behavior is described in terms of an instantaneous and equilibrium hydrostat. These functions represent the bulk response to hydrostatic loads applied over very short and very long time periods; and here a technique for determining these functions from quasi‐static and ultrasonicwave propagation experiments is presented. To illustrate the results, the instantaneous and equilibrium hydrostats are given for the solid polymer, polymethyl methacrylate.

A Layered Waveguide Technique for Determination of the Viscoelastic Properties of Liquids and Deformable Solids
View Description Hide DescriptionIn connection with the development of a technique for determining shear mechanical properties of liquids and deformable solids at high frequencies, the basic equations are presented for a layered waveguide geometry. The model is an infinite plate surrounded by an infinite viscoelastic material. The propagation of shear waves in such a system is described and the appropriate equations are developed to relate explicitly the shear mechanical properties of the surrounding material to the velocity and attenuation of the wave in the plate. For the case in which the characteristic shear mechanical impedance of the surrounding material is less than 2% of that of the plate, a simplified equation is obtained. This equation is useful for a wide range of liquids. The equations for the layered waveguide technique are compared to the corresponding equations for other techniques and the similarities are discussed.

Viscoelastic Measurements of Poly(dimethylsiloxanes) and a Polystyrene Solution Using the Layered Waveguide Technique
View Description Hide DescriptionAn experimental system is developed to measure the high frequency (1 MHz to 7 MHz) viscoelastic constants of liquids and deformable solids using a layered waveguide geometry consisting of an aluminum strip delay line upon which coatings are applied. A piezoelectric transducer mounted at one end is used to generate and receive pulses of radio frequency shear waves that reflect back and forth in the line. The change in attenuation and the phase shift of the wave in the line produced when the sample coating is applied are measured by a calibrated attenuator method and an echo superposition technique, respecively. From these measurements the viscoelastic constants of the sample are calculated using the layered waveguide equations developed earlier. The conditions under which these equations can be applied to the delay line are discussed in detail. Tests were made with a series of four poly(dimethylsiloxane) samples having static viscosities ranging from 100 to 100,000 centistokes. The measure changes are calculated for a wave traveling 1 cm in the delay line and are reproducible to within ±0.009 decibels/cm of attenuation and ±0.06 degrees/cm of phase shift in a total change of 0.5 dB/cm and 1.4 degrees/cm. The measuredviscoelastic properties agree with those reported in the literature. The delay line has a high sensitivity to liquids.Measurements on a 3.64 centipoise toluene solution of polystyrene agree with values obtained by extrapolating lower frequency measurements reported in the literature.

Time‐Dependent Stress in Polymer Melts and Review of Viscoelastic Theory
View Description Hide DescriptionNew rheogoniometer data for stress development after initiation of a steady shear are presented for three polymer melts: a low‐density polyethylene, a high‐density polyethylene, and a polystyrene. Three temperatures (160°C, 180°C, 200°C) were studied. Both the shear and normal stresses were found to “over‐shoot” at the higher shear rates. These data are compared in detail with three viscoelastic (constitutive) models: (1) the Bernstein, Kearsley, and Zapas model, using a particular functional form for the memory function; (2) the Bird‐Carreau model; and (3) the Bogue model, as modified by Bogue and White. The general features of these and certain other constitutive models are discussed. The general conclusion is that nonlinearities must be introduced wherever a time constant appears, which can be interpreted graphically as a shift of the relaxation times to lower values. Some comments about the “analogy” between shear rate and molecular weight are also made.

Anisotropic Fluid Theory: A Different Approach to the Dumbbell Theory of Dilute Polymer Solutions
View Description Hide DescriptionThe continuum theory of anisotropic fluids, as developed by Ericksen and others, has been used to formulate an expression for the time derivative of the end‐to‐end vector of a linear macromolecule. When this expression is used in conjunction with the equation describing the distribution function for a dilute solution of dumbbell elements, the results exhibit important differences from the usual dumbbell theory. Presence of an additional term in the differential equation for the distribution function leads to the prediction of both a non‐Newtonian viscosity and nonzero first and second normal stress differences in simple shearing flow. The normal stress differences are found to be of opposite sign, the secondary normal stress difference being negative. In small‐amplitude oscillatory shear flow, and in pure deformationalflow, the results are equivalent to those of the dumbbell theory. Expressions are presented for both stress and optical properties in an arbitrary homogeneous shear field.

Rheological Equations from Molecular Network Theories
View Description Hide DescriptionLodge's molecular network theories are quite successful in describing the linear viscoelastic behavior of polymer solutions and melts, but cannot account for the rate‐of‐strain dependence of various material functions. By allowing the junction‐creation rate and the probability of loss of junctions to depend on the second invariant of the rate‐of‐strain tensor, more realistic constitutive equations were obtained. Two rheological models are proposed by assuming two different mechanisms for the effect of the rate of strain on the kinetics of the network. The experimental data on three fluids (representative of eight viscoelastic fluids) are used to test the models in various flow situations. For steady simple shearing and small‐amplitude, sinusoidal simple shearing, both model A and model B are capable of fitting the four functions η, η′, and rather well over many decades of shear rate or frequency. For suddenly changing flow experiments model A is inadequate. Model B however appears to be the only rheological equation which can fit simultaneously the steady shear, complex viscosity, stress growth, and stress relaxation functions. For stress growth, the agreement with the experimental data is remarkable, especially after the other models were shown to fail drastically. Finally, an interpretation of the stress growth and relaxation phenomena is given in the light of the modified theory.

Polymers as Structured Continua
View Description Hide DescriptionThe ability of microstructure theory to predict the behavior of polymers in transient shear fields is examined. Apparent viscosity and normal stress differences are formed for the cases of shear relaxation and growth using a microstructure model. The results agree qualitatively with experimental results for three different viscoelastic solutions. The analytical results are also compared to those of the Spriggs 4‐constant model and found to be in exact agreement for low shear rates. This agreement permits determination of the phenomenological coefficients in terms of the Spriggs constants.

Significance of Viscosity Measurements in Macroscopic Suspensions of Elongated Particles
View Description Hide DescriptionThis paper investigates the conditions whereby it is possible to give an objective value to viscosity measurements of a suspension of artificial rigid fibers. It is shown that, as long as the ratio between the length of the particles and the width of the field of flow is not larger than viscosity measurements carried out with capillary viscometer and with Couette viscometer are equivalent and linearly related with the axial ratio of the particles.

Discontinuous and Dilatant Viscosity Behavior in Concentrated Suspensions. I. Observation of a Flow Instability
View Description Hide DescriptionThe rheological dilatancy of concentrated suspensions has been studied to determine the cause of this phenomenon. If monodisperse suspensions of polymeric resins are examined, dilatant viscosity behavior is transformed into a discontinuous viscosity behavior when the volume fraction of solids is raised above 0.50. Experimental evidence is presented which supports the hypothesis that the discontinuity is caused by a flow instability in which surfaces of spheres, packed in a two dimensional hexagonal packing at low shear rates, break up into less ordered arrays of spheres. Although various techniques have been used, white light diffraction from a suspension under shear provides the most dramatic evidence of the ordered packing of the spheres and the order‐disorder transition at the instability point.

Constitutive Equations for Dilute Polymer Solutions
View Description Hide DescriptionThe theories of continuum mechanics and molecular physics are brought together to provide constitutive relations for dilute polymer solutions. The required compatibility shows that the stress tensor is composed of a sum of stress tensors, each satisfying Oldroyd's differential constitutive equation. In each equation, the higher order strain rate terms are not retained, and the five material constants are found solely by utilizing results from Rouse's molecular theory. When the fluid acceleration is small, it is shown that the set of constitutive relations may be approximated by a single equation of the Oldroyd type, containing only two material constants.

The Forced Vibrating Reed: Error Analysis
View Description Hide DescriptionA linearized error analysis is performed of the forced vibrating reed test. The test variables measured are amplitude ratio, phase lag, and frequency in addition to geometric variables. Utilizing the perturbations of the complex modulus resulting from perturbations of the measured quantities enables the experimenter to design effective tests for maximum utilization of test equipment.